Overview
This lecture covers key algebra concepts for Junior Cycle Higher Level Maths: simplifying expressions, solving equations (linear and quadratic), factorizing, and managing algebraic fractions, including worked examples and exam strategies.
Simplifying Expressions
- You can only add or subtract like terms (same variables and powers), e.g., x and x, not x and x².
- If there’s no coefficient in front of a variable, assume it is 1, e.g., x = 1x.
- When adding terms with the same power, the power does not change.
- To simplify by multiplying terms, multiply coefficients first, then variables, e.g., 3 × 2y = 6y.
- When multiplying brackets, multiply each term in the bracket by the term outside; for double brackets, use distribution (FOIL).
Solving Equations
- To solve an equation, isolate the variable, performing the same operation on both sides.
- For linear equations: get variable terms on one side, constants on the other, then solve.
- For quadratic equations: use the quadratic formula (−b ± √(b²−4ac))/(2a) when the equation has x², x, and a constant.
- Quadratics always have two solutions, which may be left in decimal or surd (root) form as required.
Factorizing
- Four types:
- Highest Common Factor (HCF): factor out the biggest number and variable common to all terms.
- Grouping: only for four terms; factor in pairs, then factor the result.
- Quadratic (Trinomial): for three terms, use factor pairs method to break up the middle term, then factor in pairs.
- Difference of Two Squares: two terms, both perfect squares separated by a minus (e.g., a²−b² = (a+b)(a−b)).
Algebraic Fractions
- To simplify a single algebraic fraction, factorize numerator and denominator, then cancel common factors.
- For adding/subtracting fractions, find the lowest common denominator, adjust numerators, and combine.
- When solving equations with fractions, make all denominators the same, then equate numerators and solve.
- When denominators involve variables or complex brackets, multiply through to find a common denominator.
Exam Strategies & Example Problems
- Always show full working to get partial credit.
- Use the quadratic formula for solving quadratics and for solving fraction equations when necessary.
- Practice with past exam questions covering factorizing, simplifying fractions, and solving equations.
Key Terms & Definitions
- Like Terms — terms with identical variables and exponents that can be combined.
- Coefficient — the numerical part multiplying a variable.
- Quadratic Equation — an equation of the form ax² + bx + c = 0.
- Factorizing — expressing an expression as a product of its factors.
- Lowest Common Denominator (LCD) — the smallest expression both denominators divide into.
Action Items / Next Steps
- Practice simplifying, multiplying, and solving algebraic expressions and equations.
- Memorize recognition methods for the four types of factorization.
- Try the sample exam question at the end of the lecture and check against solutions.
- Review how to use your calculator for the quadratic formula.